Question

How do you express `0.0001/0.04020` as a decimal?

Answer 1

`1/402`

Explanation:

Take `0.0001/0.04020` and multiply top and bottom by 10000.
`{0.0001 xx 10000}/{0.04020 xx 10000}.`
Use the "move the decimal" rule. ie. `3.345 xx 100=334.5` to get:
`1/402.` This is the answer in fraction form.

If the goal was to covert the decimal directly to fractions and then solve, in `0.0001`, the `1` is in the ten thousandth column, making it the fraction `1/10000` and the 2 in 0.0402 is also in the ten thousandth column so `0.0402=402/10000`.

`0.0001/0.04020= {1/10000}/{402/10000} =1/10000-:402/10000`
`=1/10000 xx 10000/402 =1/402`.

Answer 2

Multiply numerator and denominator by `10^4` to get `1/402`, then long divide to get:

`1/402 = 0.0bar(0)2487562189054726368159203980099bar(5)`

Explanation:

To calculate `0.0001 / 0.04020` first multiply both numerator and denominator by `10^4` to get `1/402`

Assuming we want a decimal expansion of the quotient, let's use long division.

First write out the multiples of `402` we will use:

`0:color(white)(XX000)0`
`1:color(white)(XX0)402`
`2:color(white)(XX0)804`
`3:color(white)(XX)1206`
`4:color(white)(XX)1608`
`5:color(white)(XX)2010`
`6:color(white)(XX)2412`
`7:color(white)(XX)2814`
`8:color(white)(XX)3216`
`9:color(white)(XX)3618`

Then our long division starts:

Write the dividend `1.000` under the bar and the divisor `402` to the left. Since `402` is somewhat less than `1`, there are several zeros for the quotient before it 'gets going'. Once we have brought down 3 `0`'s from the dividend our initial running remainder is `1000` and the first non-zero digit of the quotient is `color(blue)(2)` resulting in `2 xx 402 = 804` to be subtracted from the remainder to yield the next remainder.

Bring down another `0` from the dividend alongside the remainder `196` to give `1960` and choose the next digit `color(blue)(4)` for the quotient, etc.

Notice that with the running remainder having arrived at `10` we are essentially back to dividing `1/402` again - that is we have found the recurring decimal expansion:

`1/402 = 0.0bar(0)2487562189054726368159203980099bar(5)`

Answer 3

I want to capitalize on George C. answer and give my version of `1/402`!!!

Explanation:

have a look:

Answer 4

Just for fun I thought I would add a solution as well. I am going to considerably limit the number of decimal places!!

`color(blue)( 0.0001/(0.04020)" "~=" "0.00024)`

Explanation:

Given:`" " 0.0001/(0.04020)`

`color(purple)("Making them into more mentally manageable numbers")` `color(purple)("and apply a correction at the end!")`

Multiply the numerator by `10^7` giving: 1000 so the correction is`xx10^(-7)`

so `0.0001/(0.04020)" "=" "1000/0.0402xx10^(-7)`

Multiply the denominator by `10^4` in the form of

`1/0.0402xx1/10^4 ->1/402` so the correction for this bit is `xx10^4`

Putting this all together gives:

`1000/402 xx (10^(4-7))" "=" "1000/402color(green)(xx10^(-3))`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

`color(blue)("Step 1")`
`" "underline(" ")`
Write as:`" " 402|1000`

Consider just the hundreds: `10-:4=2+"Remainder"`
Do not worry about the remainder!

`" "underline(" 2 ")`
Now write:`" " 402|1000`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Step 2")`

`2xx402=color(brown)(804)`

`" "underline(" 2 ")`
Now write:`" " 402|1000`
`" "color(brown)(underline(804 -))`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Step 3")`

subtract the 804 from the 1000
`" "underline(" 2 ")`
`" " 402|1000`
`" "color(brown)(underline(804 -))`
`" "196`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Step 5")`

402 > 196 so put a decimal place to the right of the 2 and put a
`color(red)(0)` to the right of 196

`" "underline(" 2"color(red)(.)" " )`
`" " 402|1000`
`" "underline(804 -)`
`" "196color(red)(0)`

`402xx5=2010 >1960` so too big
`402xx4=color(magenta)(1608)<1960` so we pick this one

so `1960-:402=color(green)(4) +"Remainder"`

So now we write:

`" "underline(" "2"."color(green)(4)" " )`
`" " 402|1000`
`" "underline(804 -)`
`" "1960`
`" "underline(color(magenta)(1608-))`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Step 6")`

`" "underline(" "2"."color(green)(4)" " )`
`" " 402|1000`
`" "underline(804 -)`
`" "1960`
`" "underline(1608-)`
`" "352`

352 < 402 so put `color(red)(0)` to the right of 352 and we repeat step 5. This cycle go on for ever if the number is irrational!
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

So far we have 2.4. Applying the correction this becomes:

`2.4 color(green)(xx10^(-3))" "->" "2.4/1000" "=" "0.00024`

`0.0001/(0.04020)" "~=" "0.00024`

Look at the beginning to see where `color(green)(xx10^(-3))` comes from.